In this work we are concerned with one of the simplest extensions of SM with a complex singlet scalar. In the model, the \(U(1)\)  symmetry is softly broken by a cubic term to \(\mathbb {Z}_3\)  symmetry (see also recent works [1]}, [2]}).
The real part of the complex scalar acquires a VEV, while the imaginary component (pseudo-Goldstone boson)
plays the role of DM candidate due to a residual \(\mathbb {Z}_2\)  symmetry in the scalar potential.
As shown in a previous study [3]}, such a cubic term could generally induce a potential barrier at tree level and trigger a strong first-order EWPT.
We find that in the two-step EWPT scenario and for the heavy scalar mass falling in the range of \(1-2\)  TeV, the model with a Higgs mixing angle satisfying \(\left| \theta \right|\gtrsim 0.2\)  (\(11.5^{\circ }\) ) can induce a strong first-order EWPT and the stochastic gravitational wave (GW) signals produced from the phase transition are potentially discoverable by future space-based interferometers, such as LISA [4]}, DECIGO [5]} and BBO [6]}.
